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Properties

Label 2-637-13.3-c1-0-21
Degree $2$
Conductor $637$
Sign $0.929 + 0.367i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.651 + 1.12i)2-s + (−1.44 − 2.49i)3-s + (0.151 − 0.262i)4-s + 2.88·5-s + (1.87 − 3.25i)6-s + 3·8-s + (−2.65 + 4.59i)9-s + (1.87 + 3.25i)10-s + (2.95 + 5.11i)11-s − 0.872·12-s + (3.31 − 1.41i)13-s + (−4.15 − 7.19i)15-s + (1.65 + 2.86i)16-s + (0.436 − 0.755i)17-s − 6.90·18-s + (−1.44 + 2.49i)19-s + ⋯
L(s)  = 1  + (0.460 + 0.797i)2-s + (−0.831 − 1.44i)3-s + (0.0756 − 0.131i)4-s + 1.28·5-s + (0.766 − 1.32i)6-s + 1.06·8-s + (−0.883 + 1.53i)9-s + (0.593 + 1.02i)10-s + (0.890 + 1.54i)11-s − 0.251·12-s + (0.920 − 0.391i)13-s + (−1.07 − 1.85i)15-s + (0.412 + 0.715i)16-s + (0.105 − 0.183i)17-s − 1.62·18-s + (−0.330 + 0.572i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.367i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.929 + 0.367i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.929 + 0.367i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.06157 - 0.392923i\)
\(L(\frac12)\) \(\approx\) \(2.06157 - 0.392923i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + (-3.31 + 1.41i)T \)
good2 \( 1 + (-0.651 - 1.12i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.44 + 2.49i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 2.88T + 5T^{2} \)
11 \( 1 + (-2.95 - 5.11i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.436 + 0.755i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.44 - 2.49i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.30 + 5.72i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.651 + 1.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.872T + 31T^{2} \)
37 \( 1 + (0.697 + 1.20i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.75 + 6.50i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.75 - 4.77i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.3T + 47T^{2} \)
53 \( 1 - 9.60T + 53T^{2} \)
59 \( 1 + (3.31 - 5.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.88 - 4.99i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2 - 3.46i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.76T + 73T^{2} \)
79 \( 1 - 0.605T + 79T^{2} \)
83 \( 1 + 6.63T + 83T^{2} \)
89 \( 1 + (4.32 + 7.48i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.88 + 6.73i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.46344422606782093832379282463, −9.883570963730732950160131776171, −8.477942970091084892346464790960, −7.38829721481833646304024647544, −6.68090350237705047870992930143, −6.14814871362150971342404653086, −5.55846951184410142523558076742, −4.43859017682249900036757608598, −2.05412434316914290697350386237, −1.42916608395552063693088818392, 1.55673566926245904266820283336, 3.25949850134240769253431019742, 3.88842867203087388304476290008, 5.01489422580397478501053742466, 5.89690958828907832784607756917, 6.53220128857062587791015124815, 8.403298490088127199232885187676, 9.288721389444393278189238305972, 10.01837025929979170651104630248, 10.76624756588184151882206790942

Graph of the $Z$-function along the critical line